Issue 
Wuhan Univ. J. Nat. Sci.
Volume 27, Number 2, April 2022



Page(s)  99  103  
DOI  https://doi.org/10.1051/wujns/2022272099  
Published online  20 May 2022 
Mathematics
CLC number: O175.2
Regularity and Energy Conservation of the Nonhomogeneous Incompressible Ideal Magnetohydrodynamics Equations
^{1}
College of Science, China Three Gorges University, Yichang
443002, Hubei, China
^{2}
College of Science & Three Gorges Mathematical Research Center, China Three Gorges University, Yichang
443002, Hubei, China
^{†} To whom correspondence should be addressed. Email: zhyp5208@163.com
Received:
24
December
2021
In this paper, we study the regularity and local energy equation of the weak solutions for nonhomogeneous incompressible ideal magnetohydrodynamics system. The conditions given on the regularity of solutions guarantee the energy to be conserved. The main method we have employed relies on the commutator estimates.
Key words: ideal magnetohydrodynamics (MHD) system / regularity / energy conservation
Biography: XIONG Jiajia, female, Master candidate, research direction: partial differential equations. Email: 752346766@qq.com
Foundation item: Supported by the National Natural Science Foundation of China (11871305, 11901346)
© Wuhan University 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
0 Introduction
The goal of this paper is to study the relationship between the regularity and energy conservation of the nonhomogeneous incompressible ideal magnetohydrodynamics (MHD) equations, reading as:(1) (2) (3) (4)where ρ, u and B represent density, speed and the magnetic field, respectively. The permeability μ is a constant, and the pressure P is unknown. We consider this system in the threedimensional periodic domain T ^{3}.
When B=0, system (1)(4) become the well known nonhomogeneous incompressible Euler equations. Onsager^{[1]} asserted that the velocity of Hölder continuity with exponential δ>1/3 guarantees energy conservation, and many researchers also studied the energy conservation of the Euler equations, which can be referred to Refs.[24] etc.
For the homogeneous ideal MHD equations, Caflisch, Klapper and Steele^{[5]} proved the conservation of energy in the periodic domain by extending the results of the Euler equations to the ideal MHD equations. Then Kang and Lee^{[6]} proved the conservation of energy and crosshelicity of the ideal MHD equations. Subsequently, Yu^{[7]} improved the previous results by using the special structure of the nonlinear term in the ideal MHD equations.
For the nonhomogeneous MHD equations, some results have also been obtained recently, which could be referred to Refs.[813] etc. Among these documents, Bie et al ^{[8]} studied the compressible MHD equations, and they gave two sufficient conditions on the regularity of solutions to ensure energy conservation. Wu et al ^{[9]} considered the incompressible MHD equations, and they proved that the regularity of the solution is sufficient to guarantee the balance of the total energy in the Besov space. However the magnetic field in Ref. [9] needs to satisfya natural question then is whether this condition is required or not.
Inspired by Refs. [3, 4, 8] and [9], we study the energy conservation of nonhomogeneous incompressible ideal MHD system (1)(4) in the threedimensional periodic domain T ^{3}. Our strategy relies on commutator estimates similar to those employed by Constantin et al ^{[4]}. In order to prove the energy conservation of system (1)(4), we first mollify the equation (1) and then select the test function ϕ to obtain an equation that approximates the energy equation. For the treatment of the nonlinear term in this equation, we use the condition to transfer the derivative to the test function ϕ by partial integral formula, which makes the regularity assumption about curlB in Ref.[9] unnecessary.
We briefly recall the definitions of Lebesgue space and Besov space , where or . Lebesgue whereAnd , wherehere .
The main result of this paper is stated as follows.
Theorem 1 Let be a weak solution to system (1)(4). Assumefor some and such that(5)Then the energy is locally conserved in the sense of distribution, that is,(6)
Remark 1 If B=0, this system reduces to the nonhomogeneous incompressible Euler equations, and our result could recover the one for the incompressible Euler equations^{[3]}.
1 Preliminaries
In this section we introduce some properties of Besov space . Let for d=3 or d=4 (according to the choice of Ω) be a standard mollifying kernel and setwith the notation . For any function ω, is welldefined on
Referring to Ref. [1], we have the following facts about functions in the Besov spaces:(7)and(8)
Moreover, is an algebra, that is the product of two functions in this space is again contained in the space.
2 Proof of Theorem 1
By smoothing (1) in space and time, we obtain(9)Taking , multiplying (9) by and integrating on , we get(10)here we take small enough so that . We can rewrite (10), using appropriate commutators, as(11)where
In order to prove our theorem, the first step is to show that the terms in the left side of (11) converge to the ones of the equation (6). For the first three terms of the left side of (11), we refer to the proof of Ref. [3]( Theorem 3.1), and get(12) (13) (14)And for the fourth term, we first introduce the following equality about curl,By smoothing (3) and multiplying it by , we obtainThen(15)
Thus, combining (11), (12), (13), (14) and (15), we find(16)To prove our result, it suffices to show , , as . For the treatment of and , we refer to the proof of Theorem 3.1 of Ref. [3], and get(17)and(18)here we request . In fact when , we know andwhich yields that . As for , in view of the equalitywe have(19)Here can be estimated asBecause the estimates of I_{1} and I_{2} are the same, we only calculate I_{1}. Firstly we observe that
By using Hölder inequality and Fubini theorem, we estimate I_{1} as(20)and(21)The estimate for is similar to that of I_{1}. Then(22)Collecting all the above estimates and putting them into (16), we obtainwhich completes the proof of Theorem 1..
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